Stability of solutions to nonlinear wave equations with switching time delay

Abstract

In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show that, under suitable conditions on the feedback operators, asymptotic stability results are available. Concrete examples included in our setting are illustrated. We give also stability results for an abstract model with alternate positive-negative damping, without delay.2000 Mathematics Subject Classification: 35L05, 93D15

Energy decay for evolution equations with delay feedbacks

“…The arguments of [19,20] could be easily extended to 𝑘(⋅) ∈ 𝐿 ∞ with sufficiently small ‖𝑘‖ ∞ . Recently, motivated by some applications, 𝐿 1 𝑙𝑜𝑐 damping coefficients 𝑘 have been considered, for instance of intermittent type (see [11,15,23]), and stability estimates have been obtained for some particular models. Here, our aim is to give a well-posedness result and an exponential decay estimate for the general model (1.1) with a damping coefficient 𝑘 belonging only to 𝐿 1 𝑙𝑜𝑐 .…”

Stability Results for Abstract Evolution Equations with Intermittent Time-Delay Feedback

“…More recently Nicaise and Pignotti [23,24] considered second-order evolution equations with intermittent delay feedback. These results have been improved and extended to some semilinear equations in [9]. In the studied models, when the delay term (which possess a destabilizing effect) is not present, a not-delayed damping acts.…”

“…On the contrary, in previous stability results for damped wave equation and intermittent delay feedback (see e.g. [24,9]) the set ω 2 has to be a subset of ω 1 . On the other hand, now the standard (not delayed) frictional damping is always present in time while in the quoted papers it is on-off like the delay feedback and it acts only on the complementary time intervals with respect to this one.…”

On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay

“…In 1976, Kirchhoff developed an equation describing the vibrations produced by a fixed series at its end, since it is considered a generalization of the d'Alembert equation, and it belongs to the wave equation models. Over time, many researchers and authors addressed these issues and problems with their continuous and rapid development, for example, see [1][2][3][4].…”